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Spectral theory for perturbed Krein Laplacians in nonsmooth domains. (English) Zbl 1191.35188

The aim of this paper is spectral properties of the Krein-von Neumann extension \(H_{K,\Omega}\) of the perturbed Laplacian \(-\Delta +V\) defined on \(C_0^\infty(\Omega)\), \(V\) is measurable, bounded and nonnegative in a bounded open set \(\Omega \subset\mathbb R^n\). A briefly definition of the Krein-von Neumann extension of appropriate \(L^2(\Omega;d^nx)\)-realizations of the differential operator \(\mathcal A\) of order \(2m,m\in\mathbb N\), is the following
\[ {\mathcal A} = \sum_{0 \leq |\alpha| \leq 2m } a_\alpha (.) D^\alpha, \]
\[ {\mathcal D}^\alpha = (-i\partial /\partial x_1)^{\alpha_1}\dots (-i\partial /\partial x_n)^{\alpha_n}, \quad \alpha=(\alpha_1,\dots,\alpha_n)\in\mathbb N_0^n, \]
\[ a_\alpha (.) \in C^\infty(\overline{\Omega}), \quad C^\infty(\overline{\Omega}) = \bigcap_{k \in N_0} C^k(\overline{\Omega}), \]
where \(\Omega \subset\mathbb R^n\) is a bounded \(C^\infty\) domain. The \(L^2(\Omega;d^nx)\)-realization \(A_{c,\Omega}\) of \({\mathcal A}\) is defined in the following way
\[ A_{c,\Omega} u = {\mathcal A} u, \quad u\in \text{dom}(A_{c,\Omega}):= C_0^\infty(\Omega). \]
If the coefficients \(a_\alpha\) are chosen such that \(A_{c,\Omega}\) is symmetric then holds
\[ (u,A_{c,\Omega}v)_{L^2(\Omega;d^nx)} = (A_{c,\Omega}u,v)_{L^2(\Omega;d^nx}), \quad u,v \in C_0^\infty (\Omega). \]
The main result of this paper is the following Weyl asymptotic formula
\[ \#\{j \in N| \lambda_{K,\Omega,j} \leq \lambda \} = (2\pi)^{-n}v_n|\Omega|\lambda^{n/2}+ O(\lambda^{(n-(1/2))/2}) \quad \text{as } \lambda \to \infty, \]
where \(v_n\) denotes the volume of the unit ball in \(\mathbb R^n\) and \(\lambda_{K,\Omega,j}\), \(j\in N\), are the eigenvalues of \(H_{K,\Omega}\). The basic idea of the proof is to use that the problem is spectrally equivalent to the buckling of a clamped plate problem. Furthermore, an abstract result of V. A. Kozlov, Spectral Asymptotic Behavior, in [J. Sov. Math. 35, 2180–2193 (1986); translation from Probl. Mat. Anal. 9, 34–56 (1984; Zbl 0632.47014)] has been used. The paper contains more than 170 references and is an essential contribution.

MSC:

35P05 General topics in linear spectral theory for PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis

Citations:

Zbl 0632.47014
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References:

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